Stress isotropic hardening

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

Powder Yield Loci For a given shear step, as the applied shear stress is increased, the powder will reach a maximum sustainable shear stress "U, at which point it yields or flows. The functional relationship between this limit of shear stress "U and applied normal load a is referred to as a yield locus, i.e., a locus of yield stresses that may result in powder failure beyond its elastic limit. This functional relationship can be quite complex for powders, as illustrated in both principal stress space and shear versus normal stress in Fig. 21-36. See Nadia (loc. cit.), Stanley-Wood (loc. cit.), and Nedderman (loc. cit.) for details. Only the most basic features for isotropic hardening of the yield surface are mentioned here. 

The stress-strain relationship for FGMs is assumed to be a bilinear form and can be described by the isotropic hardening model and kinematic hardening model as ... 

Strains and stresses were computed for the joined specimen cooled uniformly to room temperature from an assumed stress-free elevated temperature using numerical models described in detail previously [19, 20]. The coordinate system and an example of the finite element mesh utilized are shown in Figure 3. Elastie-plastic response was permitted in both the Ni and Al203-Ni composite materials a von Mises yield condition and isotropic hardening were assumed. 

Here, s means the stress tensor, a and fi are the backstress-tensors of the yield surface / and the bounding surface F, respectively. Y is the radius of the yield surface B and R are the initial size of the isotropic hardening. 

An example of a material model based on the physics of material behavior is classical metals plasticity theory. This theory, often referred to as /2-flow theory, is based on a Mises yield surface with an associated flow rule, followed by rate-independent isotropic hardening (Khan and Huang 1995). Physically, plastic flow in metals is a result of dislocation motion, a mechanism known to be driven by shear stresses and to be insensitive to hydrostatic pressure. 

The ore material is elasto-plastic with isotropic hardening mle. The plastic properties are described by yield stress Rp = 9.946 q MPa and fracture stress Rm = 12.661 MPa, which are given by bounded histograms, see Figures 6 and 7. 

Fig. 3.28. Illustration of strain hardening using a stress-strain curve. Here, we assume isotropic hardening which is explained in the text...

If yielding of the material is governed by the von Mises yield criterion, we find the yield criterion for isotropic hardening, using the changing flow stress... 

A simple isotropic hardening law can be written down for the case of linear hardening, defined by the flow stress increasing linearly with the plastic strain. Its rate formulation is [65]. 

The stress-strain diagram of a material with isotropic hardening that is deformed by uniaxial tension first and uniaxial compression afterwards can be found in figure 3.30(b). In compression, the material yields at a stress —crpi, given by the absolute value of the maximum stress in tension, api. 

If we deform a kinematically hardening material in uniaxial tension and compression, its behaviour differs drastically from the isotropically hardening material discussed above (figure 3.32). Upon load reversal, the material yields at a stress yield surface remains unchanged. In the extreme case, this may lead to plastic deformation while the stress is still tensile (figure 3.32(b)). 

Fig. 7.17. Variation of normalized equi-biaxial film stress as a function of normalized temperature, for the example considered in Section 7.5.2 where the film material is modeled as an isotropically hardening solid. The solid lines denote the response obtained from the numerical integration of (7.65) for the first three thermal cycles where it is assumed that the material properties do not vary with temperature, over the range considered. The dashed lines denote the corresponding behavior for the case where the thin film plastic response is taken to be temperature-dependent.

An ideal elastic-plastic solid is one that deforms elastically until the yield stress is reached and thereafter deforms plastically under that stress without hardening. One such material is isotropic, with Young s modulus 80 X 10 Pa and yield stress 12 X 10 Pa. Aright circular cylinder of this material, initially 0.2 m long and of radius... 

Unified equations that couple rate-independent plasticity and creep [114] are not readily available for SOFC materials. The data in the hterature allows a simple description that arbitrarily separates the two contributions. In the case of isotropic hardening FEM tools for structural analysis conveniently accept data in the form of tabular data that describes the plastic strain-stress relation for uniaxial loading. This approach suffers limitations, in terms of maximum allowed strain, typically 10 %, predictions in the behaviour during cycling and validity for stress states characterised by large rotations of the principal axes. 

The more rigorous stress/strain nonlinear material model, oflen referred to as the plastic zone method, is theoretically capable of handling any general cross section Both isotropic and kinematic hardening rules are usually available. This method is... 

In FCC materials, there are 12 different slip systems, which can contribute to the deformation process. Dislocation density histories at a peak stress of 4.5 GPa for [001], [111] and [Oil] orientations and isotropic case with [001] orientation are calculated and plotted as shown in Fig. 13. It is clear that the dislocation density is very sensitive to crystal orientation with the highest density exhibited by [111] orientation followed by the isotropic media, [011] and [001] orientations respectively. This may be attributed to the number of slip systems activated and to the way in which these systems interact. The [001] orientation has the highest symmetry among all orientations with four possible slip planes 111 that have identical Schmid factor of 0.4082, which leads to immediate work hardening. The [011] orientation is also exhibits symmetry with 2 possible slip planes that have Schmid factor of 0.4082. 

During isotropic loading, plastic deformation takes place when the isotropic stress p reaches the preconsolidation pressure pf. The pressure pf is a measure of the size of the yield surface on the isotropic axis and can be viewed as an hardening/softening parameter (the specific shape of the yield surface is described in the next section). An essential feature of the proposed model is the decrease of pf with respect to an increase in contaminant concentration. This can be expressed as... 

This relation contains two competing terms the first term represents plastic hardening as a function of the volumetric part of plastic strain, the second term describes chemical softening due to an increase in contaminant concentration. Let us consider the plastic response to an increase in contaminant concentration at constant isotropic stress. The condition p =pc =0 in equation (7) implies... 

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